foreword
note on the second printing
preface
1 in the ashes of the ether: differential topology
1. smooth manifolds
2. submanifolds
3. fiber bundles
4. tangent vectors, bundles, and fields
5. differential forms
2 looking for the forest-in the leaves: foliations
1. integral curves
2. distributions
3. integrability conditions
4. the frobenius theorem
5. the frobenius theorem in terms of differential forms
6. foliations
7. leaf holonomy
8. simple foliations
3 the fundamental theorem of calculus
1. the maurer-cartan form
2. lie algebras
3. structural equation
4. adjoint action
5. the darboux derivative
6. the fundamental theorem: local version
7. the fundamental theorem: global version
8. monodromy and completeness
4 shapes fantastic: klein geometries
1. examples of planar klein geometries
2. principal bundles: characterization and reduction
3. klein geometries
4. a fundamental property
5. the tangent bundle of a klein geometry
6. the meteor tracking problem
7. the gauge view of klein geometries
5 shapes high fantastical: cartan geometries
1. the base definition of cartan geometries
2. the principal bundle hidden in a cartan geometry
3. the bundle definition of a cartan geometry
4. development, geometric orientation, and holonomy
5. flat cartan geometries and uniformization
6. cartan space forms
7. symmetric spaces
6 riemannian geometry
1. the model euclidean space
2. euclidean and riemannian geometry
3. the equivalence problem for riemannian metrics
4. riemannian space forms
5. subgeometry of a riemannian geometry
6. isoparametric submanifolds
7 msbius geometry
1. the msbius and weyl models
2. msbius and weyl geometries
3. equivalence problems for a conformal metric
4. submanifolds of msbius geometry
5. immersed curves
6. immersed surfaces
8 projective geometry
1. the projective model
2. projective cartan geometries
3. the geometry of geodesics
4. the projective connection in a riemannian geometry
5. a brief tour of projective geometry
a ehresmann connections
1. the geometric origin of ehresmann connections
2. the reductive case
3. ehresmann connections generalize cartan connections
4. covariant derivative
b rolling without slipping or twisting
1. rolling maps
2. the existence and uniqueness of rolling maps
3. relation to levi-civita and normal connections
4. transitivity of rolling without slipping or twisting
c classification of one-dimensional effective klein pairs
1. classification of one-dimensional effective klein pairs
d differential operators obtained from symmetry
1. real representations of so2(r)
2. operators on riemannian surfaces
e characterization of principal bundles
bibliography
index
